The 64-year-old former Scotland full-back has been a member of the Hoops' backroom staff since 1997 and was promoted to first-team coach by manager Neil Lennon in October 2012.
In a statement released on their website, the club said: "Celtic Football Club confirmed today that club legend and current first-team coach Danny McGrain has experienced a minor heart attack.
"Danny is recovering well and is scheduled to be released from hospital on Friday. Clearly, the thoughts and best wishes of everyone at the club are very much with Danny as we wish him a full and speedy recovery."
McGrain won eight Scottish league titles with the Hoops between 1971 and 1986 and was capped 62 times by Scotland.
The player has had to overcome several health issues during his playing career.
Early in his career, he was diagnosed with diabetes, but with on-going medical treatment was able to play on.
McGrain encountered another major obstacle in his career when a mystery foot injury ruled him out from playing for nearly 18 months in the late 1970s, causing him to miss Scotland's World Cup campaign in Argentina.
Rangers Football Club has sent its best wishes to McGrain.